Stability and Sensitive Analysis of a Model with Delay Quorum Sensing

and Applied Analysis 3 Theorem 3. If R 0 > 1, the bacteria-free equilibrium E 0 is asymptotically stable, while if R 0 < 1, E 0 is unstable. Theproofs forTheorems 2 and 3 are trivial, so omit them. In the sequel, we study the stability of the positive equilibrium E j . First of all, we transfer it to the origin and get


Introduction
Quorum sensing is a process that enables bacteria to communicate using secreted signaling molecules called autoinducers [1].It makes bacteria regulate their gene expression collectively and control their behaviors on community scale.Quorum sensing was initially observed in the marine bacterium Vibrio fischeri about 30 years ago [2,3].Now, many other species are observed to exhibit quorum sensing behavior, including major human pathogens such as Staphylococcus aureus and Pseudomonas aeruginosa.Quorum sensing has received more and more attention (see [4][5][6][7][8][9][10][11][12][13][14][15] and the references therein) and some models are formulated to investigate its effect on the transmission of disease.Braselton and Waltman [4] formulated the dynamically allocated inhibitor production.Dockery and Keener [5] were devoted to developing and studying an ODE and a PDE mathematical models for quorum sensing in Pseudomonas aeruginosa and found that quorum sensing works because of a biochemical switch between two stable steady solutions, one with low levels of autoinducer and one with high levels of autoinducer.Koerber et al. [6] presented a mathematical model for the early stages of the infection process by Pseudomonas aeruginosa in burn wounds which accounts for the quorum sensing and the diffusion of signalling molecules in the burn-wound environment, and the effects of important parameters on the dynamic properties of the model are discussed in detail.They gave some sufficient conditions for the global asymptotic stability of two boundary equilibria which, respectively, correspond to the survival of the allelochemical producer species or the susceptible one.Fergola et al. [7] formulated an allelopathic competition model in which a distributed delay term simulates quorum sensing which regulates the delay production process of allelochemicals, and they proved the unique existence of the positive solution and the stability of biologically meaningful steady-state solutions.Anguige et al. [9] constructed a multiphase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm to investigate the effect of antiquorum sensing and antibiotic treatments on the exopolysaccharide concentration, signal level, bacterial numbers, and biofilm growth rate.The above articles leave out the immune response to the bacterial invasion.However, the immune status of the hosts has a significant impact on the transmission of an infection in a population.Literatures [11][12][13][14][15] employed a quadratic function to describe the quorum sensing of bacteria and formulate some models to characterize the competition between bacteria and the immune system, in which the existence of periodical solution, chaotic motion, and subharmonic bifurcation, the properties of Hopf bifurcation, and the stability of equilibrium et al. were investigated.As a novelty of this paper, a cubic function is used to express quorum sensing.This makes

Model Formulation
We denote by   () the concentration of the uninfected target cells,   () the concentration of the infected target cells, () the concentration of the bacteria,   () the concentration of the innate cells, and   () the concentration of the adaptive cells.The dynamic relations among them are as follows: the uninfected target cells have a natural turnover   and a halflife    and they are infected by bacteria with mass-action term  1   ; the infected target cells are cleared by halflife    or adaptive immune cells with mass action term  2     ; both the innate and the adaptive immune cells have a source term and a half-life time; for the innate immunity, the source term    includes a wide range of cells involved in the first wave of defense of the host such as natural killer cells, polymorphonuclear cells, macrophages, and dendritic cells, and for the adaptive immunity, the source term    represents the memory cells, derived from a previous infection or vaccination, a zero source means the first infection with this pathogen and there are no memory cells; both of the two kinds of cells are increased by the signals captured by the bacteria load; the bacteria population has a net growth term represented by a logistic function  20 (1 − /) and it is cleared by the innate immunity with mass action term  3   .Here, we use a function to formulize the bacteria that compete with the immune cells at time , which receive signal molecules  time units ago. 0 is a positive constant,  20 is the growth rate of bacteria, and  is the effective carrying capacity of the environment.Consequently, the vital dynamics are governed by ( Remark 1.The first equation of system (2) suggests that bacteria are controlled and increased by quorum sensing except for their net growth and they are cleared by the innate immune cells.The second equation of system (2) characterizes the dynamics of the uninfected target cells, and the third one reflects the dynamics of the infected target cells.
The uninfected target cells are infected by bacteria in mass action law and they have their own constant input flow, and the infected target cells are killed by the adaptive immune cells.The last two equations of system (2) show that each kind of the immune cells has a special source term and their responses are enhanced by the bacteria load.The target cells and the immune cells have their own half-life terms.

The Existence and Stability of Equilibria
We introduce ( Theorem 3. If  0 > 1, the bacteria-free equilibrium  0 is asymptotically stable, while if  0 < 1,  0 is unstable.
The proofs for Theorems 2 and 3 are trivial, so omit them.
In the sequel, we study the stability of the positive equilibrium   .First of all, we transfer it to the origin and get  ()  =  () +  ( − ) +  ( ()) , where The characteristic equation of ( 5) at the origin is where Equation ( 7) has two negative roots , and the other roots can be obtained by solving the following equation: It can be seen that Theorem 4. For  = 0, both  1 and  2 are unstable when they exist.
Clearly, the left side of ( 9) is continuous in  and has roots with positive real parts if and only if it has purely imaginary roots.We will determine whether (9) has purely imaginary roots or not, from which we then will be able to get conditions for all eigenvalues to have negative real parts.
Denote the eigenvalue of the characteristic equation ( 5) by  = () + (), where (), () continually depend on the delay .Under the same conditions as Theorem 5, we have (0) < 0. Since  is continuous in , one still has () < 0 and  3 remains stable if  is sufficiently small.If there exists a positive value  0 satisfying ( 0 ) = 0 , that is,  = ( 0 ) is a purely imaginary root of (9), then  3 loses its stability and eventually becomes unstable when () becomes positive.On the other hand, if such a ( 0 ) does not exist  3 is always stable.

Normal Forms on the Center Manifold
From the discussions in the above section, it can be seen that the Jacobi matrix at  4 has a uniquely simple zero eigenvalue if  1 = 1 and  ̸ = ( 2 +  2 )/ 3 .To determine the dynamic properties of  4 , we have to compute the normal forms on the center manifold.The method used is based on the center manifold reduction and normal form theory due to Faria and Magalhaes; see [16,17].

By means of 𝑅
where It is seen from ( 19) that  0 < 0 if    and  0 are small enough, which means there exists a unique positive  3 solving (18).Denote it by  and define  3 as where  is a small parameter.Obviously,  1 = 1 if  = 0. Next, we transfer  4 to the origin by Normalizing the delay by  → /, denoting   () by   (), and neglecting the higher order terms ( 2 ), we have  ()  =  ()  () +  ()  ( − 1) +  ( () , ) , ( where where From the Riesz representation theorem the linear map  can be expressed in integral form as follows: where   is a bounded variation matrix-valued function on It is well known that the eigenvalues of () with zero real parts play an important role in the bifurcation theory of RFDES.Denote (0) by  0 , and let Λ 0 = { ∈ ( 0 ) | Re = 0}.We have Λ 0 = {0}.
Using the formal adjoint theory for FDEs in [18], the phase space C can be decomposed by Λ 0 as C =  ⊕ , where  is the generalized eigenspace associated with the eigenvalues in Λ 0 ,  = { ∈ C | ⟨, ⟩ = 0 for all  ∈  * }, and the dual space  * is the generalized eigenspace for  * (0) associated with the eigenvalues in Λ 0 .Assume that Φ and Ψ are the respective dual bases of  and  * and satisfy ⟨Ψ(), Φ()⟩ = 1.We might as well choose Φ and Ψ as follows: where ⋅  is the transpose of ⋅ and  = 1/((1 ).Let  = 0.Then, the following equations hold simultaneously: As shown in [16,17], an appropriate phase space for considering normal forms of (24) is the Banach space BC of functions from [−1, 0] into R 5 , which are uniformly continuous on [−1, 0) and with a jump discontinuity at 0. Then, the elements of BC have the form + 0 , where  ∈ C,  ∈ R 5 , and so that BC is identified with C×R 5 with the norm Let  : BC →  denote the projection and then the decomposition C =  ⊕  yields a decomposition of BC by Λ 0 as the topological direct sum BC =  ⊕ Ker  with the property  ⊂ Ker , where  is an infinite dimensional complementary subspace of  and C as shown above.Now, we decompose   ∈ C 1 in (24) as   = Φ() + , where () ∈ R and  ∈  1 =  ∩ C 1 , C 1 is the subset of C consisting of continuously differentiable functions.Next, we rewrite (24) as follows: And, then, under the composition   = Φ() + , (24) can be decomposed as a system of ODEs in R × Ker  as follows: where ) is the restriction of  as an operator from  1 into Ker , and As for autonomous ODEs in R 5 , the normal forms are obtained by a recursive process of changes of variables.At a step , the terms of order  = 2 are computed from the terms of the same order and from the terms of lower orders already computed in previous steps.Assuming that steps of orders 2, 3, . . .,  − 1, have already been performed leads to where f ∈ ( f1  , f2  ) is the terms of order  in (, , ) after the previous transformations of variables and h.o.t stands for the higher order terms.Following the algorithm of [16,17] at step , using a change of variables of the form where , x ∈ R, , ŷ ∈  1 , and  1  : R 2 → R,  2  : R 2 →  1 are homogeneous polynomials of degree  in x and , after dropping the hats for simplification of notations, (37) can be put into the normal form where It can be verified that (24) satisfies nonresonance conditions; see [16,17].Then, the locally invariant manifold for (24) tangent to  at zero must be  = 0 and the flow on this manifold is given by 1-dimensional ODE ẋ =  +  1 2 (, 0, ) +  According to [16,17], we derive and then where  , = ( For  0 = 1,  0 is also an equilibrium with simple zero singularity.To discuss its stability, we employ the following perturbation form: where  = ( 3    /   ) + ( 4    /   ) and  is a small parameter.Then, the normal form of (24) near the bacteriafree equilibrium is as follows: Theorem 10.If  0 = 1, the bacteria-free equilibrium is unstable for any  > 0.

Sensitive Analysis
Sensitivity indices allow us to measure the relative change in a variable when a parameter changes.The normalized forward sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter.When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.Here, one adopts the following definition as described by Chitnis et al. [19].
To clear bacteria in the body, we must take measures to make  0 > 1 hold.For this end, it is important to determine how crucial each parameter is to  0 .The negative sign of the sensitivity index for  0 implies that increase in the relevant parameter leads to the decrease  in  0 .Note that 0 <  1 ,  2 < 1.The most sensitive parameter is the growth rate  20 , which has a high impact on  0 and should be targeted by intervention strategies.To reduce  20 to ensure  0 > 1, we may develop an inhibitor to control the reproduction of bacteria or to kill bacteria individual, which agrees with an intuitive expectation.
To understand how  20 affects  0 , we might as well assume  4 =  3 ,    =    , and    =    , and then gets  0 = ( 3    / 20    )(1 + ).Clearly, for  = 0, only the innate immune cell competes with bacteria.Figure 1 exhibits the relationship between  0 and  20 with  3 =    = 0.02,    = 0.04, and  = 0, 2. It can be seen from the plot that as  20 decreases  0 increases faster when  = 2 than the case of  = 0, which reveals that vaccination or other strategies adopted to stimulate immunity of the body are beneficial to the clearance of bacteria.

Conclusions
This paper formulates the competition between bacteria and immune system by DDEs.Then, the qualitative properties of the model are analyzed.Specially, by virtue of the center manifold reduction and normal form theory due to Faria and Magalhaes [16,17], the normal form of system (2) associated with zero eigenvalue is computed, from which one deduces that the bacteria-free equilibrium  0 and the positive equilibrium  4 are unstable under the conditions of  0 = 1 and  1 = 1, respectively.Next, sensitivity analysis and numerical simulations indicate that the effective reproductive rate  20 is the most sensitive parameter to  0 .Theorem 3 suggests the strategies target the decrease of the growth rate which can be successful in disease elimination.On the biological viewpoint, the terms  3    /   and  4    /   measure the respective strengths of the innate and adaptive immune system defense against the bacterial challenge, while the factor  20 measures the bacteria's offensive strength.So with ( 3    )/( 20    ) + ( 4    )/( 20    ), we can compare the strength of the immune system against the bacterial offensive.Thus, Theorems 2 and 3 have the biological explications: in the domain of attraction of  0 , bacteria will be cleared if  0 > 1; that is, the strength of the immune system defense against the bacteria challenge is not weaker than the bacteria's offensive strength; in the domain of attraction of  3 , bacteria coexist with immune cells when  0 < 1 and the bacterial challenge is weaker than bacteria's offensive strength.

the
Mathematics Tianyuan Funds of NSFC (no.11226260); the Scientific Research Plan Projects of Shaanxi Education Department (nos.12JK0851, 2013JK0611); the Key Laboratory of Simulation and Control for Population Ecology (Xinyang Normal University), Xinyang 464000, China (no.201004); the Program for Innovative Research Team of Science and Technology of University of the Henan Province (no.2010IRTSTHN006); the Innovation Scientists and Technicians Troop Construction Projects of the Henan Province.